That really depends on what you’re talking about when you say Single-Payer, and how you categorize what it does. I’d argue that your definition is, in itself, highly misleading.
Single-payer is commonly considered a universal healthcare service run by the government that provides health services for the entire nation. When defined thusly, virtually all industrialized nations have some sort of single-payer healthcare.
This does not mean that no other options are available, and it doesn’t mean that the single payer trust covers every possible service. In the UK for example, one can opt for additional private insurance and can engage private medical services. In Australia, Medicare is for everyone, but doesn’t cover every possible service, and also may require a small co-pay (unless the facility opts for bulk billing). Ambulance services, for example, are excluded from coverage. One would engage a private insurer to cover things not included in Aussie Medicare.
But the common thread is that all of these nations have a nationalized web of hospitals and services paid for by a public trust that provide basic medical services to everyone. These are all types of single payer, even if there are other options and ancillary services that are available for a fee.
The point is to provide a standard of medical care to everyone by socializing that cost, and centralizing the regulation of costs for services for what is considered a Public Good. And that’s what every advanced country does, but us. If you object to calling that ’Single-Payer’, and prefer to refer to this as ‘universal access’, I don’t think anyone really cares.
Calling it a ‘lie’, however, is simply mendacious.